Firstly, the use of debt/GDP as the ratio is somewhat arbitrary. One could easily look at debt/government revenue, and it may be a more useful metric. However, the convention to use GDP for scaling these series. (One can argue that GDP represents the full income capacity of the economy for debt service.)
For countries that borrow in a foreign currency (or in a currency where they do not directly control the central bank), a high debt ratio indicates a greater risk of default. There is no gurantee that they can their hands on the foreign currency to repay debt. You need to look at other things as well, but it's one common metric.
If the country controls the currency it borrows in, it should be able to arrange its affairs to avoid default. (There is no guarantee, however.)
However, even if we ignore default risk, the greater the debt level, the greater the interest expense for the government. The exact mechanisms are debated by various schools of thought, but it is reasonably safe to argue that greater interest payments will squeeze out other spending if we want to keep inflation around a target level. There are two ways in which a greater debt load increases interest payments.
- Even if the rate of interest is unchanged, a greater amount of debt implies more interest to pay. This effect holds by definition.
- A greater debt-to-GDP ratio might increase interest rates. The magnitude of this effect is controversial. The example of Japan in recent decades (high debt ratios, low nominal yields) demonstrates that the effect is not very large, even if it exists.
There is a school of thought that argues that we need to "pay back" debt in some sense; this is often referred to as the governmental budget constraint. This view can be summarised as: greater debt now means greater future taxes, and people do not like taxes. (This is sometimes called Ricardian Equivalence.) The exact mechanism is controversial. (I have severe doubts about the mathematics behind it, for what that's worth.)
In the case where the central bank has bought a lot of government debt (quantitative easing), the expectation is that the central bank will eventually reverse that policy. In any event, the central bank needs to pay interest on resserves if it wishes to raise market interest rates to control inflation, and the consolidated government will face a similar interest cost as the case of bill issuance.
Finally, the money supply is related to debt levels; both are forms of government liabilities. In most modelling traditions, people only target a certain anount of money holdings (that earn no interest), and the rest is invested in bonds/bills. Therefore, we only expect money to grow in line with transaction needs, and this level may have little resemblance to total debt levels.